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In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):

Let $g$ be a positive multiplicative function and let $A$ and $B$ be two constants such that for all $y\geq 1,$ $$\sum_{p\leq y} g(p) \log{p} \leq Ay \quad \textrm{ and } \quad \sum_{p}\sum_{\nu \geq 2} \frac{g(p^{\nu})\log{p^{\nu}}}{p^{\nu}}\leq B.$$ Then, for $x\geq 1,$ $$\frac{1}{x} \sum_{n \leq x} g(n)\ll (A+B+1) \prod_{p\leq x} \left(1-\frac{1}{p}+\sum_{\nu \geq 1} \frac{g(p^{\nu})}{p^{\nu}}\right),$$ where the implicit constant is absolute. Unfortunately, in the library of my university, I did not find Tenenbaum's book and I need to know the explicit value of the implicit constant in the previous result. Can someone help me? Thanks in advance.

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In the French edition, it is said that the considered constant does not exceed $4(1+9\lambda_{1}+\lambda_{1}\lambda_{2}/(2-\lambda_{2})^2)$ where $\lambda_{1}>0$, $0\leq \lambda_{2}<2$ are such that $g(p^\nu)\leq\lambda_{1}\lambda_{2}^{\nu-1}$.

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  • $\begingroup$ @ijasdhashdasd Thank you for prividing me by a good reference! $\endgroup$ – Khadija Mbarki May 16 '16 at 5:48
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You can also find a very clean explicit version of Tenenbaum's inequality as Proposition 2.10 in www.dms.umontreal.ca/~andrew/PDF/Pretentious010611.pdf . The proof is short enough that you should be able to re-produce it as a Lemma (including of course the appropriate references).

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